Super-resolution approach to novel problems in machine learning inspired by deep networks
Abstract
The recent exciting advancement of uorescence light microscopy, with the capability of viewing well below the hundredth nanometer scale for studying molecular activities, was coined super-resolution" by the biology and bio-chemistry communities. In mathematics, the model of super-resolution can be formulated as some unknown linear combination f(x; t) of Dirac delta distributions (x(t) ÀÀ xk(t)) at some point-sources xk(t) in the q-dimensional Euclidean space Rq, q 2, with corresponding coecients ak(t), k = 1; ;K, where t denotes the time parameter. The challenge is the inverse problem of resolving the blind-source" f(x; t) by determining the number K of point- sources xk(t), their positions in Rq, together with their corresponding light intensities" ak(t), at any time-instant t. This mathematical model indeed addresses the problem of super-resolution in uorescence light microscopy, since the point-sources xk(t) are allowed to be as close to one another as desired, under certain appropriate conditions. Important applications to bio-medical and biology research include: pre-cancer diagnosis by monitoring culture of organ tissues, obtained from needle biopsies or surgery; timely diagnosis of the metastasis process, i.e. break-away cells from an initial tumor to spread to other sites in the human body; and observation of RNA movement from their sites of chromosomal synthesis to their functional locations in a human cell for eukaryotic gene read-out. While Fourier transform and Fourier series have been the most popular starting point in the recent mathematics literature for resolving the super-resolution problem, the PIs believe that the Gaussian-convolution approach, introduced and developed in their recently completed ARO project W911NF-15-1-0385, is more useful for real-world applications, such as those in bio-medical and biology, as discussed above, for at least two reasons. Firstly, while the Fourier approach maps the time-domain to the frequency-domain, our Gaussian-convolution approach preserves both the spatial information and motion activities of the point-sources. Secondly, all point-sources in real-world applications have positive physical sizes, as opposed to mathematical points that are represented by Dirac delta distributions. Therefore, it is more productive to formulate the problem of super-resolution as the inverse problem of resolving a general blind source measure (x) from its convolution with some general kernel, including the Gaussian. One of the proposed problems in this proposal is to extend our previous results to resolve the the blind-source of point-masses represented by Gaussian kernels with di erent variances. On the other hand, the advent and spectacular success (and some recently observed shortcomings) of deep networks has given rise to the need of a fresh look at the fundamental concepts in machine learning. In classical machine learning, the unknown functional relationship in the data is modeled as a convolution of a kernel with the marginal distribution , and therefore the goal is to resolve the measure , from given values of this convolution at randomly selected points, which can be considered as a blind source - essentially the same goal as the problem of the mathematical problem of super-resolution, except for some nuances such as the nature of the data. This insight leads to a study of super-resolution problems and machine learning problems in a uni ed manner, where the new mathematical tools to be developed in our proposed research project for one set of problems are also useful for the other set of problems. This research will result in mathematically well-founded constructions of deep and shallow networks that are robust and able to learn with a minimal number of training data, with theoretical guarantees on the reliability of their outputs. Success of the proposed uni ed approach is not limited to the advancement of machine learning and
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- W911NF2110218
Entities
People
- Hrushikesh Mhaskar
Organizations
- Army Contracting Command
- Claremont Graduate University
- United States Army